Editing Strategic voting (section)

=== Myerson–Weber strategy ===
An example of a rational voter strategy is described by [[Roger Myerson|Myerson]] and Weber.<ref>{{cite journal |last1=Myerson |first1=Roger B. |last2=Weber |first2=Robert J. |year=1993 |title=A Theory of Voting Equilibria |url=http://www.kellogg.northwestern.edu/research/math/papers/782.pdf |journal=The American Political Science Review |volume=87 |issue=1 |pages=102–114 |doi=10.2307/2938959 |jstor=2938959 |s2cid=143828854 |hdl-access=free |hdl=10419/221141}}</ref> The strategy is broadly applicable to a number of single-winner voting methods that are additive point methods, such as [[Plurality voting method|Plurality]], [[Borda count|Borda]], [[Approval voting|Approval]], and [[Range voting|Range]]. The strategy is optimal in the sense that the strategy will maximize the voter's [[Utility#Expected utility|expected utility]] when the number of voters is sufficiently large.

This rational voter model assumes that the voter's utility of the election result is dependent only on which candidate wins and not on any other aspect of the election, for example showing support for a losing candidate in the vote tallies. The model also assumes the voter chooses how to vote individually and not in collaboration with other voters.

Given a set of ''k'' candidates and a voter let:

: ''v<sub>i</sub>'' = the number of points to be voted for candidate ''i''
: ''u<sub>i</sub>'' = the voter's gain in utility if candidate ''i'' wins the election
: ''p<sub>ij</sub>'' = the (voter's perceived) ''pivot probability'' that candidates ''i'' and ''j'' will be tied for the most total points to win the election.

Then the voter's ''prospective rating'' for a candidate ''i'' is defined as:

: <math>R_i = \sum_{j \neq i} \; p_{ij} \cdot (u_i - u_j)\,</math>

The gain in expected utility for a given vote is given by:

: <math>G(p,v,u) = \sum_{i=1}^k \; v_i \cdot R_i\,</math>

The gain in expected utility can be maximized by choosing a vote with suitable values of ''v<sub>i</sub>'', depending on the voting method and the voter's prospective ratings for each candidate. For specific voting methods, the gain can be maximized using the following rules:

* Plurality: Vote for the candidate with the highest prospective rating. This is different from choosing the best of the frontrunners, which is a common [[heuristic]] approach to voting. In rare cases, the highest prospective rating can belong to a weak candidate (one with a low probability of winning).
* Borda: Rank the candidates in decreasing order of prospective rating.
* Approval: Vote for all candidates that have a positive prospective rating.
* Range: Vote the maximum (minimum) for all candidates with a positive (negative) prospective rating.

Pivot probabilities are rarely estimated in [[political forecasting]], but can be estimated from predicted winning probabilities. An important special case occurs when the voter has no information about how other voters will vote. This is sometimes referred to as the ''zero information strategy''. In this special case, the ''p<sub>ij</sub>'' pivot probabilities are all equal and the rules for the specific voting methods become:

* Plurality: Vote for the most preferred (highest utility) candidate. This is the sincere plurality vote.
* Borda: Rank the candidates in decreasing order preference (decreasing order of utility). This is the sincere ranking of the candidates.
* Approval: Calculate the average utility of all candidates. Vote for all candidates that have a higher-than-average utility; do not vote for any candidates that have a lower-than-average utility.
* Range: Calculate the average utility of all candidates. Vote the maximum points for all candidates that have a higher-than-average utility; vote the minimum points for all candidates that have a lower-than-average utility; vote any value for a candidate with a utility equal to the average.

Myerson and Weber also describe voting equilibria that require all voters use the optimal strategy and all voters share a common set of ''p<sub>ij</sub>'' pivot probabilities. Because of these additional requirements, such equilibria may in practice be less widely applicable than the strategies.